Question

True or False? Justify your answer with a proof or a counterexample. The rectangular coordinates of the point \(\left(4, \frac{5 \pi}{6}\right)\) are \((2 \sqrt{3},-2)\).

Short answer

False. The correct rectangular coordinates are (-2\sqrt{3}, 2).

Step-by-step solution

Understand the Problem

The point given is in polar coordinates: \( (r, \theta) = (4, \frac{5 \pi}{6}) \). We need to convert these polar coordinates into rectangular coordinates (Cartesian coordinates) to verify if they equal \((2 \sqrt{3}, -2)\).

Convert Polar to Rectangular Coordinates

The conversion from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\) involves two formulas:1. \( x = r \cdot \cos(\theta) \)2. \( y = r \cdot \sin(\theta) \).Given \( r = 4 \) and \( \theta = \frac{5\pi}{6}\), we will use these formulas to find \( x \) and \( y \).

Calculate the x-Coordinate

Using the formula \( x = r \cdot \cos(\theta) \), substitute \( r = 4 \) and \( \theta = \frac{5\pi}{6} \):\[ x = 4 \cdot \cos\left(\frac{5\pi}{6}\right) \].Since \( \cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \), the calculation becomes:\[ x = 4 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3} \].

Calculate the y-Coordinate

Using the formula \( y = r \cdot \sin(\theta) \), substitute \( r = 4 \) and \( \theta = \frac{5\pi}{6} \):\[ y = 4 \cdot \sin\left(\frac{5\pi}{6}\right) \].Since \( \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \), the calculation becomes:\[ y = 4 \cdot \frac{1}{2} = 2 \].

Compare Calculated Coordinates with Given Coordinates

Compare the calculated rectangular coordinates \((-2\sqrt{3}, 2)\) with the given coordinates \((2\sqrt{3}, -2)\). They do not match, indicating the statement is false.

Key concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance from a fixed point and an angle from a fixed direction. The fixed point is known as the pole, and the fixed direction is typically the positive x-axis in standard polar coordinates. This system uses pairs in the form

• \((r, \theta)\),
• where \(r\) is the radius - the line from the pole - and \(\theta\) is the angle measured in radians.

To locate a point in polar coordinates, you simply move \(r\) units away from the pole and rotate the direction specified by \(\theta\). This system shines when dealing with circular or rotational symmetry due to its radial nature. If you find terms like negative \(\theta\) or \(r\) confusing, remember that:

• A negative angle \(\theta\) often means rotating clockwise.
• A negative \(r\) suggests moving in the opposite direction of the angle.

Polar coordinates are particularly useful in fields like physics and engineering, where angles and radial distances come into play frequently.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point's location using two values: one for the horizontal axis (usually called \(x\)) and one for the vertical axis (usually called \(y\)). This system is straightforward and widely used due to its simplicity. To understand this system, think of the plane as divided by an x-axis (horizontal) and a y-axis (vertical). Any point is represented by an ordered pair \((x, y)\), where:

• \(x\) reflects the horizontal distance from the origin.
• \(y\) denotes the vertical distance from the origin.

This coordinate system is particularly advantageous for graphing linear equations, placing special emphasis on intersections of axes to determine positions. Despite its widespread use, it can be less intuitive when dealing with rotational or circular motion.

Coordinate Transformation

Coordinate transformation refers to the process of converting one set of coordinates to another. In the context of polar and rectangular coordinates, this involves mapping points from a circular system to a grid-like system and vice versa. The conversion formulas are rooted in basic trigonometric relationships.

• Polar to Rectangular:
  To convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas:
  • \(x = r \cdot \cos(\theta)\)
  • \(y = r \cdot \sin(\theta)\)

The purpose of these transformations is not just for practical conversion, but also to help visualize geometry or physics problems in the way that makes them easiest to solve. Understanding both coordinate systems' strengths allows choosing the most efficient method for problem-solving scenarios.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles and ratios of triangle sides. They form the backbone of converting polar coordinates to rectangular coordinates.

• Key Functions:
  • Sine (\(\sin\)) provides the ratio of the length of the opposite side to the hypotenuse in a right triangle.
  • Cosine (\(\cos\)) gives the ratio of the adjacent side's length to the hypotenuse.

In coordinate transformation:

• The \(\cos\) function helps determine the x-coordinate.
• The \(\sin\) function finds the y-coordinate.

Both functions are periodic and oscillate between -1 and 1, important for periodic phenomena like waves. Mastery of these functions is essential for understanding how angles influence positions in the coordinate plane.